Citation/Abstract

In this dissertation we study a one-vertex minimal triangulation of a genus two handlebody. We describe the normal planar surfaces in this triangulation. The process of layering on additional tetrahedra gives us a method to construct Heegaard splittings of closed three-manifolds. We give sufficient conditions on the layers to give a triangulated 3-manifold with a strongly irreducible Heegaard splitting of genus two that is 0-efficient.

A 3-manifold with a one-vertex triangulation allows us to represent the normal surfaces using the vertex-linking surface. We project the elementary disks of a normal surface onto the vertex-linking surface and study the tracks created by the projected boundaries of the elementary disks.